mentioned features of the circulation developed already after several iterations of the minimizer. We conclude that using proper initialization cannot alone make assimilation of climatology successful. Some additional constraints are required.
0 200 400 600 800 1000
10−2 10−1 100
iteration
cost function
advection diffusion data
Figure 5.11: Evolution of contributions into the cost function in assimilation of WOA94 climatology starting from the forward estimate. The dot, dashed and dash-dot lines show the contributions from the departure of the model density from climatological density, the advection and the diffusion parts of the equation for the potential density, respectively.
5.6 Constraint on the deep pressure gradient
The inability to get a reasonable circulation pattern by applying the stationary inversion procedure is perhaps not surprising in the light of conclusions made by Tziperman et al. (1992b) and the indication of multiple local equilibria that follows from the results presented by Marotzke and Wunsch (1993). The question is, however, how to proceed. The recommendation of Marotzke (1992) and then by Tziperman et al. (1992a,b) is to impose the steadiness constraint by penalizing differences between the initial and final state separated by some finiteinterval of time. This is not a very cheap way to proceed as time intervals used for this purpose are from about half a year to five years, and many such time intervals are to be integrated with both forward and adjoint model in the process of minimization. The idea, therefore is, to try to understand what
76 CHAPTER 5. NORTH ATLANTIC CIRCULATION is the difference between optimized Levitus and forward estimate circulations and design an additional constraint that would direct the solution in a ‘proper’
way.
To begin, let us return to the estimate of the circulation obtained from the forward model. Its circulation in the upper ocean does not fully match our a priori knowledge about the circulation in the North Atlantic. One of the examples is the Gulf Stream going far to the north, which is a feature shared with many other models of similar or even better resolution. Therefore we would not like to simply penalize the misfit between velocities of the forward and optimized solution. However we are satisfied with the integral properties of the forward estimate and with its deep ocean circulation showing a well-formed Deep Western Boundary Current. The deep circulation is driven by deep pressure gradients, and this suggests to compare the pattern of deep pressure anomalies in solutions discussed thus far. Figure 5.12 presents, from top to bottom, pressure anomalies at the depth 2000 m in the Levitus analyzed, Levitus optimized and the forward solutions. First we note that while the analyzed and optimized pressures differ slightly in the position and the strength of the DWBC, they share the common wrong feature close to the eastern coast.
Both demonstrate the presence of northward current there, and optimization only strengthens it.
Further analysis shows that the minimization procedure is mostly acting through the changes in the barotropic velocity close to the eastern coast.
Namely this destroys the integral properties of the circulation.
The deep pressure anomaly in the forward run (see the bottom panel of Fig. 5.12) differs essentially from analyzed and optimized pressures. It shows the noticeably narrower DWBC, and more important, there is no northward current along the eastern coast.
One would be interested in minimization that leaves deep ocean circulation intact while changing it (by adjusting density) in the upper ocean. The moti-vation here comes from recognizing the fact that the density field and, hence, the pressure field are dynamically consistent (with the velocity field) in the forward estimate (obtained by running FEOM over 10 years starting from the Levitus climatology) while they are not necessarily consistent in the analyzed and optimized stationary solutions.
In order to keep deep ocean pressure close to the forward estimate we introduce a constraint on the deep pressure gradient. We want the gradient of full pressure of the optimal solution be close to the one of the forward estimate over some range of depths
∇P +gρ0∇ζ → ∇Pf e+gρ0∇ζf e, (5.3) where Pf e and ζf e are the hydrostatic pressure and sea surface height given by the forward estimate, respectively, and P and ζ are those of the optimal solution, ρ0 is the mean density.
5.6. CONSTRAINT ON THE DEEP PRESSURE GRADIENT 77 We use this weak constraint below 2000 m depth by adding a new term to the cost function J:
J7 = Z
Ω
Z
Ω
(∇P +gρ0∇ζ− ∇Pf g −gρ0∇ζf g)WP(x, y, z, x0y0z0) (∇P +gρ0∇ζ− ∇Pf g −gρ0∇ζf g)dΩdΩ0 (5.4) The weights are set to zero above 2000 m. Thus we allow the upper ocean (above 2000 m) to change while keeping the lower ocean circulation close to the forward estimate.
In order to see how much we should displace the structure of the density field from the climatology in order to approach the forward estimate pressure we plot in Fig.5.13 the difference
(Pf e+gρ0ζf e−PL−gρ0ζL) gρ0
, (5.5)
at the depth of 2000 m. HerePLis the hydrostatic pressure computed from the Levitus climatology, ζL is the sea surface height elevation which corresponds to Levitus climatology analysis (see section 5.2).
Figure 5.13 shows that along the DWBC the change inζL of approximately 0.1 m is required in order to have pressure gradient equal to the forward estimate at 2000 m depth. This corresponds to approximately 0.05 kg/m3 change in the density over upper 2000 m or 1 kg/m3 if the change in density occurs only in the upper 100 m. Obviously, those changes are pretty big and would not always let us approach the data within standard deviations.
We should mention that the gradient of pressure is computed in finite element sense, i.e. its values at nodes are weighted with the test function over element clusters
∇P +gρ0∇ζ → Z
Ω
(∇P +gρ0∇ζ)˜udΩ. (5.6) This form of defining the pressure gradient is consistent with the definition of other terms which are included into the cost function (see section 5.4).
Similar to weighting the residuals of potential density equation, we assume that volume weighting is sufficient to mask information on variance of the deep pressure gradient at all locations when defining its weights and take the same variance everywhere. Its magnitude is estimated by evaluating gradients of the difference shown in Fig. 5.13.
In the following chapters we conduct assimilation of different climatologies using constraint on the deep pressure gradient (see table 5.1).
78 CHAPTER 5. NORTH ATLANTIC CIRCULATION
Figure 5.12: Pressure anomalies at 2000 m depth corresponding to the di-agnostic of the Levitus climatology (upper panel), to the optimized solution based on Levitus data (section 5.5 (second panel) and to the forward estimate (lower panel).